Equations — Hard Practice Quiz

A Algebra cheat sheet for Equations — every key formula with its symbols defined — plus a hard-level practice quiz to test recall.

Formulas & key concepts

Linear Equation

$$ax + b = 0 \Rightarrow x = -\frac{b}{a}$$

Quadratic Formula

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Discriminant: If \(D > 0\), 2 real roots. If \(D = 0\), 1 real root. If \(D < 0\), 2 complex roots.

$$D = b^2 - 4ac$$

Vieta's Formulas (Quadratic)

$$x_1 + x_2 = -\frac{b}{a}, \quad x_1 x_2 = \frac{c}{a}$$

Practice quiz

  1. A quadratic equation $x^2 + px + q = 0$ has roots $x_1$ and $x_2$. If one root is twice the other (i.e., $x_1 = 2x_2$), express $p^2$ in terms of $q$.

    • $p^2 = \frac{9q}{2}$
    • $p^2 = \frac{2q}{9}$
    • $p^2 = \frac{3q}{2}$
    • $p^2 = 4q$

    Answer: $p^2 = \frac{9q}{2}$

  2. For what value of $k$ does the quadratic equation $x^2 + (k+1)x + k = 0$ have exactly one real root?

    • $k=1$
    • $k=0$
    • $k=-1$
    • $k=2$

    Answer: $k=1$

  3. If $x_1$ and $x_2$ are the roots of $ax^2 + bx + c = 0$, what is the quadratic equation whose roots are $x_1+k$ and $x_2+k$?

    • $ax^2 + (b - 2ak)x + (c - bk + ak^2) = 0$
    • $ax^2 + (b + 2ak)x + (c + bk + ak^2) = 0$
    • $ax^2 + (b - ak)x + (c - bk + k^2) = 0$
    • $ax^2 + (b + ak)x + (c + bk + k^2) = 0$

    Answer: $ax^2 + (b - 2ak)x + (c - bk + ak^2) = 0$

  4. Consider the quadratic equation $ax^2 + bx + c = 0$ where $a > 0$. If $b^2 - 4ac > 0$, how does increasing the value of $c$ (while $a$ and $b$ remain constant) affect the real roots?

    • The real roots move closer to each other, potentially becoming one real root or no real roots.
    • The real roots move further apart.
    • The real roots remain unchanged.
    • The number of real roots always increases.

    Answer: The real roots move closer to each other, potentially becoming one real root or no real roots.

  5. For the quadratic equation $3x^2 - 7x + 2 = 0$, find the sum of the squares of its roots. Round to two decimal places.

    • 4.11
    • 5.44
    • 1.33
    • 2.33

    Answer: 4.11

  6. Consider the system of equations: $y = 2x - 1$ and $y = x^2 - 3x + 5$. Find the sum of the $x$-coordinates of the intersection points.

    • $5$
    • $6$
    • $-5$
    • $1$

    Answer: $5$

  7. For what range of values of $k$ does the quadratic equation $x^2 - 2kx + (k^2 - 1) = 0$ have two distinct real roots?

    • All real values of $k$
    • $k > 1$
    • $k < -1$
    • No real values of $k$

    Answer: All real values of $k$

  8. If $x_1$ and $x_2$ are the roots of the quadratic equation $x^2 - 5x + 6 = 0$, find the value of $\frac{1}{x_1} + \frac{1}{x_2}$.

    • $\frac{5}{6}$
    • $\frac{6}{5}$
    • $\frac{1}{6}$
    • $\frac{1}{5}$

    Answer: $\frac{5}{6}$

  9. A quadratic equation $2x^2 - 5x + k = 0$ has one root that is twice the other. Find the value of $k$. Round to two decimal places.

    • 2.78
    • 1.39
    • 0.83
    • 2.50

    Answer: 2.78

  10. If the quadratic equation $x^2 + px + q = 0$ has two distinct real roots, and $p$ is positive, what can be concluded about the sign of $q$?

    • $q$ can be positive, negative, or zero.
    • $q$ must be positive.
    • $q$ must be negative.
    • $q$ must be zero.

    Answer: $q$ can be positive, negative, or zero.

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